I'm in the process of converting all the images (mostly photographs) on my two academic sites to PNGs, and have spent the last week studying PNG stuff (though whether I've learned much is another question). So here's a summary from a one-week-expert (i.e., amateur), and what I decided to do:
The PNG format does support transparency. In practice, some browsers still can't display this aspect of PNGs, though, so if you have, say, a round image that depends on a transparent background to blend in, then some users will see an ugly non-transparent rectangle. So the few images I have like this I have left as GIFs for now.
PNG is a lossless format, like TIFF, whereas JPG is a lossy format. This makes PNG good for storing master copies of images; I'm now scanning original photos in PNG format for storage; JPG should definitely not be used for this purpose. Even if you decide to display in JPG, keep the PNG original and you will always be able to recreate anything from it.
Loss-less-ness was one of the virtues of TIFF, and it was why people often stored archival scans in TIFF format. PNG can replace this, and in principle it can store the files more compactly (certainly not any bigger than TIFF).
I said in principle, because although PNGs can compress losslessly very well, early editing software (such as early Photoshop; not sure about current versions) had lousy compression algorithms. (That's lousy, not lossy.) :) I'm using Photoshop 4 on a Mac, and my PNGs of photographs are about 25% bigger than corresponding GIFs, and much bigger than JPGs. This is *not* inherent in the PNG format: it is just poor implementation of compression in Photoshop. If I resave those same files in a newer version, they may shrink considerably.
In spite of this size difference, I am now converting all my photo images to PNGs and am scanning new images as PNGs. Why? Because the truly amazing feature of PNG is that it can preserve the *gamma* information of the image with the file. I'm not a graphics specialist so can't explain the technicalities, but you can think of gamma information as information about the brightness and contrast of the original, so that when the image is called up *on any browser* it is automatically color-corrected and displays beautifully.
This last feature so outweighs anything else as far as photos are concerned that once I learned about it enough to understand what was going on I immediately started converting everything. I'm a Mac user, and I also take natural history photos. When I've put them up on my machine they look good, and then I look at them on a PC and they look like mud. As a Mac person, I'd just insert the obligatory expletive about PCs and assume that's just the way things would always be. Then I discovered PNG gamma correction (last week). My images now have just the same color balance and brightness on a PC screen that they have on my Mac.
Since this is of interest to many people I think, I've made a demo page on my personal site that shows this effect with three sample images. If the moderator will permit: just take my username and add .net/server/pngtest.html and you will see.
A starting point to learn more is the W3C PNG page: